Mathematische Annalen

, Volume 355, Issue 4, pp 1469–1525

Sur la densité des représentations cristallines de \(\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p)\)



Let \({\mathfrak X }_d\) be the \(p\)-adic analytic space classifying the semisimple continuous representations \(\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p) \rightarrow \mathrm GL _d(\overline{\mathbb Q }_p)\). We show that the crystalline representations are Zarski-dense in many irreducible components of \({\mathfrak X }_d\), including the components made of residually irreducible representations. This extends to any dimension \(d\) previous results of Colmez and Kisin for \(d = 2\). For this we construct an analogue of the infinite fern of Gouvêa–Mazur in this context, based on a study of analytic families of trianguline \((\varphi ,\Gamma )\)-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline \((\varphi ,\Gamma )\)-modules, as well as the density of the crystalline \((\varphi ,\Gamma )\)-modules in this family. These results may be viewed as a local analogue of the theory of \(p\)-adic families of finite slope automorphic forms and they are new already in dimension \(2\). The technical heart of the paper is a collection of results about the Fontaine–Herr cohomology of families of trianguline \((\varphi ,\Gamma )\)-modules.

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.C.N.R.S, Centre de Mathématiques Laurent Schwartz, École Polytechnique Palaiseau CedexFrance

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